The Erdös–Pósa property for matroid circuits

The number of disjoint cocircuits in a matroid is bounded by its rank. There are, however, matroids with arbitrarily large rank that do not contain two disjoint cocircuits; consider, for example, M ( K n ) and U n , 2 n . Also the bicircular matroids B ( K n ) have arbitrarily large rank and have no 3 disjoint cocircuits. We prove that for each k and n there exists a constant c such that, if M is a matroid with rank at least c, then either M has k disjoint cocircuits or M contains a U n , 2 n -, M ( K n ) -, or B ( K n ) -minor.